3.652 \(\int \frac{\sqrt{2+3 x}}{a+b x^2} \, dx\)

Optimal. Leaf size=427 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right )}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right )}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}-\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}} \]

[Out]

(3*ArcTanh[(Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]] - Sqrt[2]*b^(1/4)*Sqrt[2 + 3*x])/Sqrt[2*Sqrt[b] - Sqrt[9*a + 4*b
]]])/(Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] - Sqrt[9*a + 4*b]]) - (3*ArcTanh[(Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]] + Sqr
t[2]*b^(1/4)*Sqrt[2 + 3*x])/Sqrt[2*Sqrt[b] - Sqrt[9*a + 4*b]]])/(Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] - Sqrt[9*a + 4
*b]]) + (3*Log[Sqrt[9*a + 4*b] - Sqrt[2]*b^(1/4)*Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]]*Sqrt[2 + 3*x] + Sqrt[b]*(2
+ 3*x)])/(2*Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]]) - (3*Log[Sqrt[9*a + 4*b] + Sqrt[2]*b^(1/4)*Sqrt
[2*Sqrt[b] + Sqrt[9*a + 4*b]]*Sqrt[2 + 3*x] + Sqrt[b]*(2 + 3*x)])/(2*Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] + Sqrt[9*a
+ 4*b]])

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Rubi [A]  time = 0.530846, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {700, 1129, 634, 618, 206, 628} \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right )}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right )}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}-\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + 3*x]/(a + b*x^2),x]

[Out]

(3*ArcTanh[(Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]] - Sqrt[2]*b^(1/4)*Sqrt[2 + 3*x])/Sqrt[2*Sqrt[b] - Sqrt[9*a + 4*b
]]])/(Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] - Sqrt[9*a + 4*b]]) - (3*ArcTanh[(Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]] + Sqr
t[2]*b^(1/4)*Sqrt[2 + 3*x])/Sqrt[2*Sqrt[b] - Sqrt[9*a + 4*b]]])/(Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] - Sqrt[9*a + 4
*b]]) + (3*Log[Sqrt[9*a + 4*b] - Sqrt[2]*b^(1/4)*Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]]*Sqrt[2 + 3*x] + Sqrt[b]*(2
+ 3*x)])/(2*Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]]) - (3*Log[Sqrt[9*a + 4*b] + Sqrt[2]*b^(1/4)*Sqrt
[2*Sqrt[b] + Sqrt[9*a + 4*b]]*Sqrt[2 + 3*x] + Sqrt[b]*(2 + 3*x)])/(2*Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] + Sqrt[9*a
+ 4*b]])

Rule 700

Int[Sqrt[(d_) + (e_.)*(x_)]/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2 + a*e^2 - 2*c*d
*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1129

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/
c, 2]}, Dist[1/(2*c*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[1/(2*c*r), Int[x^(m - 1)/(q + r*x + x^2),
x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{2+3 x}}{a+b x^2} \, dx &=6 \operatorname{Subst}\left (\int \frac{x^2}{9 a+4 b-4 b x^2+b x^4} \, dx,x,\sqrt{2+3 x}\right )\\ &=\frac{3 \operatorname{Subst}\left (\int \frac{x}{\frac{\sqrt{9 a+4 b}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{2+3 x}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}-\frac{3 \operatorname{Subst}\left (\int \frac{x}{\frac{\sqrt{9 a+4 b}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{2+3 x}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}\\ &=\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{9 a+4 b}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{2+3 x}\right )}{2 b}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{9 a+4 b}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{2+3 x}\right )}{2 b}+\frac{3 \operatorname{Subst}\left (\int \frac{-\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}{\sqrt [4]{b}}+2 x}{\frac{\sqrt{9 a+4 b}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{2+3 x}\right )}{2 \sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}-\frac{3 \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}{\sqrt [4]{b}}+2 x}{\frac{\sqrt{9 a+4 b}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{2+3 x}\right )}{2 \sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}\\ &=\frac{3 \log \left (\sqrt{9 a+4 b}-\sqrt{2} \sqrt [4]{b} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} \sqrt{2+3 x}+\sqrt{b} (2+3 x)\right )}{2 \sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}-\frac{3 \log \left (\sqrt{9 a+4 b}+\sqrt{2} \sqrt [4]{b} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} \sqrt{2+3 x}+\sqrt{b} (2+3 x)\right )}{2 \sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{2 \left (2-\frac{\sqrt{9 a+4 b}}{\sqrt{b}}\right )-x^2} \, dx,x,-\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}{\sqrt [4]{b}}+2 \sqrt{2+3 x}\right )}{b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{2 \left (2-\frac{\sqrt{9 a+4 b}}{\sqrt{b}}\right )-x^2} \, dx,x,\frac{\sqrt{2} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}{\sqrt [4]{b}}+2 \sqrt{2+3 x}\right )}{b}\\ &=\frac{3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} \left (\frac{\sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}{\sqrt [4]{b}}-\sqrt{2} \sqrt{2+3 x}\right )}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} \left (\frac{\sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}{\sqrt [4]{b}}+\sqrt{2} \sqrt{2+3 x}\right )}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}+\frac{3 \log \left (\sqrt{9 a+4 b}-\sqrt{2} \sqrt [4]{b} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} \sqrt{2+3 x}+\sqrt{b} (2+3 x)\right )}{2 \sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}-\frac{3 \log \left (\sqrt{9 a+4 b}+\sqrt{2} \sqrt [4]{b} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}} \sqrt{2+3 x}+\sqrt{b} (2+3 x)\right )}{2 \sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}+\sqrt{9 a+4 b}}}\\ \end{align*}

Mathematica [A]  time = 0.139488, size = 133, normalized size = 0.31 \[ \frac{\sqrt{3 \sqrt{-a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{-a}-2 \sqrt{b}}}\right )-\sqrt{3 \sqrt{-a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{-a}+2 \sqrt{b}}}\right )}{\sqrt{-a} b^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + 3*x]/(a + b*x^2),x]

[Out]

(Sqrt[3*Sqrt[-a] - 2*Sqrt[b]]*ArcTan[(b^(1/4)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[-a] - 2*Sqrt[b]]] - Sqrt[3*Sqrt[-a] +
2*Sqrt[b]]*ArcTanh[(b^(1/4)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[-a] + 2*Sqrt[b]]])/(Sqrt[-a]*b^(3/4))

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Maple [B]  time = 0.227, size = 944, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(1/2)/(b*x^2+a),x)

[Out]

1/12*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(3/2)*(9*a*b+4*b^2)^(1/2)*ln(-(2+3*x)*b^(1/2)+(2+3*x)^(1/2)*(2*(b*(
9*a+4*b))^(1/2)+4*b)^(1/2)-(9*a+4*b)^(1/2))-1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(3/2)*(9*a*b+4*b^2)^(1/2
)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2)*arctan((-2*b^(
1/2)*(2+3*x)^(1/2)+(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2))/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1
/2))-1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(1/2)*ln(-(2+3*x)*b^(1/2)+(2+3*x)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+
4*b)^(1/2)-(9*a+4*b)^(1/2))+1/3*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)/
(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2)*arctan((-2*b^(1/2)*(2+3*x)^(1/2)+(2*(b*(9*a+4*b))^
(1/2)+4*b)^(1/2))/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2))-1/12*(2*(9*a*b+4*b^2)^(1/2)+4*b
)^(1/2)/a/b^(3/2)*(9*a*b+4*b^2)^(1/2)*ln((2+3*x)*b^(1/2)+(2+3*x)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)+(9*a+
4*b)^(1/2))+1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(3/2)*(9*a*b+4*b^2)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1
/2)/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2)*arctan((2*b^(1/2)*(2+3*x)^(1/2)+(2*(b*(9*a+4*b
))^(1/2)+4*b)^(1/2))/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2))+1/6*(2*(9*a*b+4*b^2)^(1/2)+4
*b)^(1/2)/a/b^(1/2)*ln((2+3*x)*b^(1/2)+(2+3*x)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)+(9*a+4*b)^(1/2))-1/3*(2
*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9
*a+4*b))^(1/2)-4*b)^(1/2)*arctan((2*b^(1/2)*(2+3*x)^(1/2)+(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2))/(4*(9*a+4*b)^(1/2
)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 \, x + 2}}{b x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(3*x + 2)/(b*x^2 + a), x)

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Fricas [A]  time = 2.23057, size = 721, normalized size = 1.69 \begin{align*} -\frac{1}{2} \, \sqrt{-\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} + 2}{a b}} \log \left (a b^{2} \sqrt{-\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} + 2}{a b}} \sqrt{-\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) + \frac{1}{2} \, \sqrt{-\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} + 2}{a b}} \log \left (-a b^{2} \sqrt{-\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} + 2}{a b}} \sqrt{-\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) + \frac{1}{2} \, \sqrt{\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} - 2}{a b}} \log \left (a b^{2} \sqrt{\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} - 2}{a b}} \sqrt{-\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) - \frac{1}{2} \, \sqrt{\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} - 2}{a b}} \log \left (-a b^{2} \sqrt{\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} - 2}{a b}} \sqrt{-\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

-1/2*sqrt(-(3*a*b*sqrt(-1/(a*b^3)) + 2)/(a*b))*log(a*b^2*sqrt(-(3*a*b*sqrt(-1/(a*b^3)) + 2)/(a*b))*sqrt(-1/(a*
b^3)) + sqrt(3*x + 2)) + 1/2*sqrt(-(3*a*b*sqrt(-1/(a*b^3)) + 2)/(a*b))*log(-a*b^2*sqrt(-(3*a*b*sqrt(-1/(a*b^3)
) + 2)/(a*b))*sqrt(-1/(a*b^3)) + sqrt(3*x + 2)) + 1/2*sqrt((3*a*b*sqrt(-1/(a*b^3)) - 2)/(a*b))*log(a*b^2*sqrt(
(3*a*b*sqrt(-1/(a*b^3)) - 2)/(a*b))*sqrt(-1/(a*b^3)) + sqrt(3*x + 2)) - 1/2*sqrt((3*a*b*sqrt(-1/(a*b^3)) - 2)/
(a*b))*log(-a*b^2*sqrt((3*a*b*sqrt(-1/(a*b^3)) - 2)/(a*b))*sqrt(-1/(a*b^3)) + sqrt(3*x + 2))

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Sympy [A]  time = 4.2523, size = 56, normalized size = 0.13 \begin{align*} 6 \operatorname{RootSum}{\left (20736 t^{4} a^{2} b^{3} + 576 t^{2} a b^{2} + 9 a + 4 b, \left ( t \mapsto t \log{\left (576 t^{3} a b^{2} + 8 t b + \sqrt{3 x + 2} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**(1/2)/(b*x**2+a),x)

[Out]

6*RootSum(20736*_t**4*a**2*b**3 + 576*_t**2*a*b**2 + 9*a + 4*b, Lambda(_t, _t*log(576*_t**3*a*b**2 + 8*_t*b +
sqrt(3*x + 2))))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(b*x^2+a),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError